Nội dung bài giảng
Giải các bất phương trình:
a) \(\sqrt {{x^2} + x - 6} < x - 1\)
b) \(\sqrt {2x - 1} \le 2x - 3\)
c) \(\sqrt {2{x^2} - 1} > 1 - x\)
d) \(\sqrt {{x^2} - 5x - 14} \ge 2x - 1\)
Đáp án
a) Ta có:
\(\eqalign{
& \sqrt {{x^2} + x - 6} < x - 1\cr& \Leftrightarrow \left\{ \matrix{
{x^2} + x - 6 \ge 0 \hfill \cr
x - 1 > 0 \hfill \cr
{x^2} + x - 6 < {(x - 1)^2} \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
\left[ \matrix{
x \le 3 \hfill \cr
x \ge 2 \hfill \cr} \right. \hfill \cr
x > 1 \hfill \cr
3x < 7 \hfill \cr} \right. \Leftrightarrow 2 \le x < {7 \over 3} \cr} \)
Vậy \(S = {\rm{[}}2,{7 \over 3})\)
b) Ta có:
\(\eqalign{
& \sqrt {2x - 1} \le 2x - 3 \Leftrightarrow \left\{ \matrix{
2x - 1 \ge 0 \hfill \cr
2x - 3 \ge 0 \hfill \cr
2x - 1 \le {(2x - 3)^2} \hfill \cr} \right. \cr
& \Leftrightarrow \left\{ \matrix{
x \ge {1 \over 2} \hfill \cr
x \ge {3 \over 2} \hfill \cr
4{x^2} - 14x + 10 \ge 0 \hfill \cr} \right.\cr& \Leftrightarrow \left\{ \matrix{
x \ge {3 \over 2} \hfill \cr
\left[ \matrix{
x \le 1 \hfill \cr
x \ge {5 \over 2} \hfill \cr} \right. \hfill \cr} \right. \Leftrightarrow x \ge {5 \over 2} \cr} \)
Vậy \(S = {\rm{[}}{5 \over 2}; + \infty )\)
c) Ta có:
\(\eqalign{
& \sqrt {2{x^2} - 1} > 1 - x \Leftrightarrow \left[ \matrix{
\left\{ \matrix{
1 - x < 0 \hfill \cr
2{x^2} - 1 > 0 \hfill \cr} \right. \hfill \cr
\left\{ \matrix{
1 - x \ge 0 \hfill \cr
2{x^2} - 1 > {(1 - x)^2} \hfill \cr} \right. \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x > 1 \hfill \cr
\left\{ \matrix{
x \le 1 \hfill \cr
{x^2} + 2x - 2 > 0 \hfill \cr} \right. \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
x > 1 \hfill \cr
\left\{ \matrix{
x \le 1 \hfill \cr
\left[ \matrix{
x < - 1 - \sqrt 3 \hfill \cr
x > - 1 + \sqrt 3 \hfill \cr} \right. \hfill \cr} \right. \hfill \cr} \right.\cr& \Leftrightarrow \left[ \matrix{
x < - 1 - \sqrt 3 \hfill \cr
x > - 1 + \sqrt 3 \hfill \cr} \right. \cr} \)
Vậy \(S = ( - \infty , - 1 - \sqrt 3 ) \cup ( - 1 + \sqrt 3 , + \infty )\)
d) Ta có:
\(\eqalign{
& \sqrt {{x^2} - 5x - 14} \ge 2x - 1 \cr
& \Leftrightarrow \left[ \matrix{
\left\{ \matrix{
2x - 1 < 0 \hfill \cr
{x^2} - 5x - 14 \ge 0 \hfill \cr} \right. \hfill \cr
\left\{ \matrix{
2x - 1 \ge 0 \hfill \cr
{x^2} - 5x - 14 \ge {(2x - 1)^2} \hfill \cr} \right. \hfill \cr} \right.\cr& \Leftrightarrow \left[ \matrix{
\left\{ \matrix{
x < {1 \over 2} \hfill \cr
\left[ \matrix{
x \le - 2 \hfill \cr
x \ge 7 \hfill \cr} \right. \hfill \cr} \right. \hfill \cr
\left\{ \matrix{
x \ge {1 \over 2} \hfill \cr
3{x^2} + x + 15 \le 0 \hfill \cr} \right. \hfill \cr} \right. \Leftrightarrow x \le - 2 \cr} \)
Vậy \(S = (-∞, -2]\)